Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học được đăng trên tạp chí toán học quốc tế đề tài: Some results for the q-Bernoulli, q-Euler numbers and polynomials | Kim and Kim Advances in Difference Equations 2011 2011 68 http content 2011 1 68 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Some results for the q-Bernoulli q-Euler numbers and polynomials Daeyeoul Kim 1 and Min-Soo Kim2 Correspondence minsookim@ 2Department of Mathematics KAiSt 373-1 Guseong-dong Yuseong-gu Daejeon 305-701 South Korea Full list of author information is available at the end of the article Abstract The q-analogues of many well known formulas are derived by using several results of q-Bernoulli q-Euler numbers and polynomials. The q-analogues of Z-type functions are given by using generating functions of q-Bernoulli q-Euler numbers and polynomials. Finally their values at non-positive integers are also been computed. 2010 Mathematics Subject Classification 11B68 11S40 11S80. Keywords Bosonic p-adic integrals Fermionic p-adic integrals q-Bernoulli polynomials q-Euler polynomials generating functions q-analogues of -type functions q-analogues of the Dirichlet s L-functions 1. Introduction Carlitz 1 2 introduced q-analogues of the Bernoulli numbers and polynomials. From that time on these and other related subjects have been studied by various authors see . 3-10 . Many recent studies on q-analogue of the Bernoulli Euler numbers and polynomials can be found in Choi et al. 11 Kamano 3 Kim 5 6 12 Luo 7 Satoh 9 Simsek 13 14 and Tsumura 10 . For a fixed prime p Zp Qp and Cp denote the ring of Ji-adic integers the field of p-adic numbers and the completion of the algebraic closure of Qp respectively. Let p be the Ji-adic norm on Q with IpIp p-1. For convenience p will also be used to denote the extended valuation on Cp. The Bernoulli polynomials denoted by Bn x are defined as Bn x k 0 n 0 where Bk are the Bernoulli numbers given by the coefficients in the power series t tk e - 1 rZ Bkk - k 0 From the above definition we see Bk s are all rational numbers. Since e 1 1
